Numerical Methods for the Solution of Population Balance Equations Coupled with Computational Fluid Dynamics

Abstract

This review article discusses the solution of population balance equations, for the simulation of disperse multiphase systems, tightly coupled with computational fluid dynamics. Although several methods are discussed, the focus is on quadrature-based moment methods (QBMMs) with particular attention to the quadrature method of moments, the conditional quadrature method of moments, and the direct quadrature method of moments. The relationship between the population balance equation, in its generalized form, and the Euler-Euler multiphase flow models, notably the two-fluid model, is thoroughly discussed. Then the closure problem and the use of Gaussian quadratures to overcome it are analyzed. The review concludes with the presentation of numerical issues and guidelines for users of these modeling approaches.

Keywords: computational fluid dynamics; crystallization; gas–liquid systems; liquid–liquid systems; multiphase flow; population balance equation; quadrature-based moment methods.